On the Nontrivial Zeros of the Riemann Zeta Function

EasyChair Preprint no. 9139, version 5

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6 pagesDate: November 2, 2022

Abstract

The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. In 2011, Solé and Planat stated that, the Riemann hypothesis is true if and only if the inequality $\zeta(2) \cdot \prod_{p\leq x} (1+\frac{1}{p}) > e^{\gamma} \cdot \log \theta(x)$ holds for all $x \geq 5$, where $\theta(x)$ is the first Chebyshev function, $\gamma \approx 0.57721$ is the Euler-Mascheroni constant, $\zeta(x)$ is the Riemann zeta function and $\log$ is the natural logarithm. In this note, using Solé and Planat criterion, we prove that, when the Riemann hypothesis is false, then there are infinitely many natural numbers $x$ for which $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + \varepsilon \cdot \log x \leq 2.062$ could be satisfied for some $\varepsilon > 0$. Since the inequality $\frac{\log x}{\sqrt{x}} - \frac{10}{\sqrt{x}} + 2 \cdot \log x + \varepsilon \cdot \log x \leq 2.062$ never holds for every $\varepsilon > 0$ and large enough $x$, then the Riemann hypothesis is true by principle of non-contradiction.

Keyphrases: Chebyshev function, Riemann hypothesis, Riemann zeta function